If they’re a ‘don’t eat meat because bunnies are cute’ type of vegan you are probably on a hiding to nothing. If their veganism is founded in beliefs about human health, environmental impact or concerns about how farmed meat is raised, you might be able to build a case around ideas that Hunted wild meat is more ethical and lower impact than farmed meat, and maybe that as a hunter you’re morally more morally consistent and have more integrity than people who just eat chicken nuggets. As I’ve posted elsewhere I have a brother in law who eats an almost exclusively vegan diet for the latter reasons, but is enthusiastic about the occasional bit of venison or high quality meat of known provenance.
There’s a lot to be said for listening and being sympathetic - if they’re particularly polarised and unused to dealing with people in field sports their assumption will be that you’re a bloodthirsty bigoted lunatic. Just showing you’re not (I assume) will go some way to shattering their expectations. Everyone likes to be listened to, taken seriously and be treated with respect.
Unless there is a pressing need for the deer to be managed though, you might find that this is all a bit of a long shot.
Aside from that, be diplomatic, don’t push too hard, (but don’t be too soft) deflect/turn rather than confront and all the usual ‘sales’ techniques for convincing people to agree with you.
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